Absolute value
The absolute value of a real number , denoted , is the unsigned portion of . Geometrically, is the distance between and zero on the real number line.
The absolute value function exists among other contexts as well, including complex numbers.
Contents
[hide]Real numbers
When is real, is defined as For all real numbers and , we have the following properties:
- (Alternative definition)
- (Non-negativity)
- (Positive-definiteness)
- (Multiplicativeness)
- (Triangle Inequality)
- (Symmetry)
Note that
and
Complex numbers
For complex numbers , the absolute value is defined as , where and are the real and imaginary parts of , respectively. It is equivalent to the distance between and the origin, and is usually called the complex modulus.
Note that , where is the complex conjugate of .
Examples
- If , for some real number , then or .
- If , for some real numbers , , then or , and therefore or .
Problems
- Find all real values of if .
- Find all real values of if .
- (AMC 12 2000) If , where , then find .
File:Example.jpg==See Also==
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